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<h1 id="RT0-P0-Element-for-Poisson-Equation-in-2D">RT0-P0 Element for Poisson Equation in 2D<a class="anchor-link" href="#RT0-P0-Element-for-Poisson-Equation-in-2D">&#182;</a></h1>
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<p>This example is to show the rate of convergence of the linear finite element approximation of the Poisson equation on the unit square:</p>
$$- \nabla (d \nabla) u = f \; \hbox{in } (0,1)^2$$<p>for the following boundary conditions</p>
<ul>
<li>pure Dirichlet boundary condition: $u = g_D \text{ on } \partial \Omega$.</li>
<li>Pure Neumann boundary condition: $d\nabla u\cdot n=g_N \text{ on } \partial \Omega$.</li>
<li>mixed boundary condition: $u=g_D \text{ on }\Gamma_D, \nabla u\cdot n=g_N \text{ on }\Gamma_N.$</li>
</ul>
<p>Find $(\sigma , u)$ in $H_{g_N,\Gamma_N}(div,\Omega)\times L^2(\Omega)$ s.t.</p>
$$ (d^{-1}\sigma,\tau) + (div \tau, u)  = \langle \tau \cdot n, g_D \rangle_{\Gamma_D} \quad \forall \tau \in H_{0,\Gamma_N}(div,\Omega)$$$$ (div \sigma, v)                =  -(f,v) \quad \forall v \in L^2(\Omega) $$<p>where</p>
<p>$$H_{g,\Gamma}(div,\Omega) = \{\sigma \in H(div,\Omega); \sigma \cdot n = g  \text{ on } \Gamma \subset \partial\Omega \}.$$</p>
<p>The unknown $\sigma = d\nabla u$ is approximated using the lowest order Raviart-Thomas element (RT0) and $u$ by piecewise constant element (P0).</p>
<p><strong>References</strong></p>
<p><strong>Subroutines</strong>:</p>

<pre><code>- PoissonRT0
- squarePoissonRT0
- mfemPoisson
- PoissonRT0mfemrate

</code></pre>
<p>The method is implemented in <code>PoissonRT0</code> subroutine and tested in <code>squarePoissonRT0</code>. Together with other elements (BDM1), <code>mfemPoisson</code> provides a concise interface to solve Poisson equation in mixed formulation. The RT0-P0 element is tested in <code>PoissonRT0mfemrate</code>. This doc is based on <code>PoissonRT0mfemrate</code>.</p>

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<h2 id="RT0-Lowest-Order-H(div)-Element-in-2D">RT0 Lowest Order H(div) Element in 2D<a class="anchor-link" href="#RT0-Lowest-Order-H(div)-Element-in-2D">&#182;</a></h2><p>We explain degree of freedoms and basis functions for Raviart-Thomas element on a triangle.</p>
<h3 id="Asecond-orientation">Asecond orientation<a class="anchor-link" href="#Asecond-orientation">&#182;</a></h3><p>The dofs and basis depends on the orientation of the mesh. We shall use the asecond orientation, i.e., <code>elem(t,1)&lt; elem(t,2)&lt; elem(t,3)</code> not the positive orientation. Given an <code>elem</code>, the asecond orientation can be constructed by</p>

<pre><code>    [elem,bdFlag] = sortelem(elem,bdFlag);  % ascend ordering

</code></pre>
<p>Note that <code>bdFlag</code> should be sorted as well.</p>
<p>The local edge is also asecond <code>[2 3; 1 3; 1 2]</code> so that the local orientation is consistent with the global one and thus no need to deal with the sign difference when the positive oritentation is used. Read <a href="../mesh/scdoc.html">Simplicial complex in two dimensions</a> for more discussion of indexing, ordering and orientation.</p>

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<div class="prompt input_prompt">In&nbsp;[3]:</div>
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<div class=" highlight hl-matlab"><pre><span></span><span class="n">node</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">;</span> <span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">];</span>
<span class="n">elem</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">];</span>
<span class="n">edge</span> <span class="p">=</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span><span class="p">;</span> <span class="mi">1</span> <span class="mi">3</span><span class="p">;</span> <span class="mi">1</span> <span class="mi">2</span><span class="p">];</span>
<span class="n">figure</span><span class="p">;</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">findnode</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
<span class="n">findedge</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">edge</span><span class="p">,</span><span class="s">&#39;all&#39;</span><span class="p">,</span><span class="s">&#39;rotvec&#39;</span><span class="p">);</span>
</pre></div>

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<h3 id="Local-bases-of-RT0-element">Local bases of RT0 element<a class="anchor-link" href="#Local-bases-of-RT0-element">&#182;</a></h3><p>Suppose <code>[i,j]</code> is the k-th edge. The two dimensional curl is a rotated graident defined as $\nabla^{\bot} f = (-\partial_y f, \partial _x f).$ The basis of this edge along with its divergence are given by</p>
$$ \phi_k = \lambda_i \nabla^{\bot} \lambda_j - \lambda_j \nabla^{\bot} \lambda_i. $$<p>Inside one triangular, the 3 bases corresponding to 3 local edges [2 3; 1
3; 1 2] are:</p>
$$ \phi_1 = \lambda_2 \nabla^{\bot} \lambda_3 - \lambda_3 \nabla^{\bot} \lambda_2. $$<p></p>
$$ \phi_2 = \lambda_1 \nabla^{\bot} \lambda_3 - \lambda_3 \nabla^{\bot} \lambda_1. $$$$ \phi_3 = \lambda_1 \nabla^{\bot} \lambda_2 - \lambda_2 \nabla^{\bot} \lambda_1. $$<p>The dual basis is the line integral over an orientated edge</p>
$$\int_{e_i} \phi_j \cdot n_i \, ds = \delta(i,j),$$<p></p>
<p>where $n_i = t_i^{\bot}$ is the rotation of the unit tangential vector of $e_i$ by $90^{\deg}$ counterclockwise.</p>

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<h3 id="Local-to-global-index-map">Local to global index map<a class="anchor-link" href="#Local-to-global-index-map">&#182;</a></h3><p>Three local edges are <code>locEdge = [2 3; 1 3; 1 2]</code>. The pointer from the local to global index can be constructured by</p>

<pre><code>[elem2dof,edge] = dofedge(elem);</code></pre>

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<div class=" highlight hl-matlab"><pre><span></span><span class="p">[</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">]</span> <span class="p">=</span> <span class="n">squaremesh</span><span class="p">([</span><span class="mi">0</span> <span class="mi">1</span> <span class="mi">0</span> <span class="mi">1</span><span class="p">],</span> <span class="mf">0.5</span><span class="p">);</span>
<span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Dirichlet&#39;</span><span class="p">);</span>
<span class="p">[</span><span class="n">elem</span><span class="p">,</span><span class="n">bdFlag</span><span class="p">]</span> <span class="p">=</span> <span class="n">sortelem</span><span class="p">(</span><span class="n">elem</span><span class="p">,</span><span class="n">bdFlag</span><span class="p">);</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">findnode</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
<span class="n">findelem</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="p">[</span><span class="n">elem2dof</span><span class="p">,</span><span class="n">edge</span><span class="p">]</span> <span class="p">=</span> <span class="n">dofedge</span><span class="p">(</span><span class="n">elem</span><span class="p">);</span>
<span class="n">findedge</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">edge</span><span class="p">,</span><span class="s">&#39;all&#39;</span><span class="p">,</span><span class="s">&#39;rotvec&#39;</span><span class="p">);</span>
<span class="n">display</span><span class="p">(</span><span class="n">elem2dof</span><span class="p">);</span>
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<pre>
elem2dof =

  8�3 uint32 matrix

    8    3    2
   11    6    5
   15   10    9
   16   13   12
    5    3    1
    7    6    4
   12   10    8
   14   13   11

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<h2 id="Assembling-the-matrix-equation">Assembling the matrix equation<a class="anchor-link" href="#Assembling-the-matrix-equation">&#182;</a></h2><p>We discuss several issues in the assembling.</p>
<h3 id="Mass-matrix">Mass matrix<a class="anchor-link" href="#Mass-matrix">&#182;</a></h3><p>The mass matrix can be computed by</p>

<pre><code>M = getmassmatvec(elem2edge,area,Dlambda,'RT0');</code></pre>

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<h3 id="divergence-matrix">divergence matrix<a class="anchor-link" href="#divergence-matrix">&#182;</a></h3><p>The ascend ordering orientation is not consistent with the induced orientation. The second edge would be <code>[3 1]</code> for the consistent orientation. So <code>[1 -1 1]</code> is used in the construction of div operator.</p>
<p>For triangle t, the basis for the constant function space is $p = 1$, the characteristic function. So in the computation of divergence operator, <code>elemSign</code> should be used to correct the sign. In the output of <code>gradbasis</code>, <code>-Dlambda</code> is always the outwards normal direction. The signed area could be negative but in the ouput, <code>area</code> is the absolute value (for the easy of integration on elements) and <code>elemSign</code> is used to record elements with negative area.</p>

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<div class=" highlight hl-matlab"><pre><span></span><span class="p">[</span><span class="n">Dlambda</span><span class="p">,</span><span class="n">area</span><span class="p">,</span><span class="n">elemSign</span><span class="p">]</span> <span class="p">=</span> <span class="n">gradbasis</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">B</span> <span class="p">=</span> <span class="n">icdmat</span><span class="p">(</span><span class="n">double</span><span class="p">(</span><span class="n">elem2dof</span><span class="p">),</span><span class="n">elemSign</span><span class="o">*</span><span class="p">[</span><span class="mi">1</span> <span class="o">-</span><span class="mi">1</span> <span class="mi">1</span><span class="p">]);</span>
<span class="n">display</span><span class="p">(</span><span class="n">full</span><span class="p">(</span><span class="n">B</span><span class="p">))</span>
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<pre>  Columns 1 through 13

     0     1    -1     0     0     0     0     1     0     0     0     0     0
     0     0     0     0     1    -1     0     0     0     0     1     0     0
     0     0     0     0     0     0     0     0     1    -1     0     0     0
     0     0     0     0     0     0     0     0     0     0     0     1    -1
    -1     0     1     0    -1     0     0     0     0     0     0     0     0
     0     0     0    -1     0     1    -1     0     0     0     0     0     0
     0     0     0     0     0     0     0    -1     0     1     0    -1     0
     0     0     0     0     0     0     0     0     0     0    -1     0     1

  Columns 14 through 16

     0     0     0
     0     0     0
     0     1     0
     0     0     1
     0     0     0
     0     0     0
     0     0     0
    -1     0     0

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<h3 id="Boundary-edges">Boundary edges<a class="anchor-link" href="#Boundary-edges">&#182;</a></h3><p>Direction of boundary edges may not be the outwards normal direction of the domain since now <code>elem</code> is ascend orientation. <code>edgeSign</code> is introduced to record this inconsistency.</p>

<pre><code>    edgeSign = ones(NE,1);
    idx = (bdFlag(:,1) ~= 0) &amp; (elemSign == -1); % first edge is on boundary
    edgeSign(elem2edge(idx,1)) = -1;
    idx = (bdFlag(:,2) ~= 0) &amp; (elemSign == 1);  % second edge is on boundary
    edgeSign(elem2edge(idx,2)) = -1;
    idx = (bdFlag(:,3) ~= 0) &amp; (elemSign == -1); % third edge is on boundary
    edgeSign(elem2edge(idx,3)) = -1;</code></pre>

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<h2 id="Test-Examples">Test Examples<a class="anchor-link" href="#Test-Examples">&#182;</a></h2><h3 id="Mixed-boundary-condition">Mixed boundary condition<a class="anchor-link" href="#Mixed-boundary-condition">&#182;</a></h3>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">%% Setting</span>
<span class="p">[</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">]</span> <span class="p">=</span> <span class="n">squaremesh</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span><span class="mf">0.25</span><span class="p">);</span> 
<span class="n">mesh</span> <span class="p">=</span> <span class="n">struct</span><span class="p">(</span><span class="s">&#39;node&#39;</span><span class="p">,</span><span class="n">node</span><span class="p">,</span><span class="s">&#39;elem&#39;</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">option</span><span class="p">.</span><span class="n">L0</span> <span class="p">=</span> <span class="mi">1</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">maxIt</span> <span class="p">=</span> <span class="mi">4</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">printlevel</span> <span class="p">=</span> <span class="mi">1</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">elemType</span> <span class="p">=</span> <span class="s">&#39;RT0&#39;</span><span class="p">;</span>
<span class="n">pde</span> <span class="p">=</span> <span class="n">sincosNeumanndata</span><span class="p">;</span>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">%% Mix Dirichlet and Neumann boundary condition.</span>
<span class="n">option</span><span class="p">.</span><span class="n">solver</span> <span class="p">=</span> <span class="s">&#39;uzawapcg&#39;</span><span class="p">;</span>
<span class="n">mesh</span><span class="p">.</span><span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Dirichlet&#39;</span><span class="p">,</span><span class="s">&#39;~(x==0)&#39;</span><span class="p">,</span><span class="s">&#39;Neumann&#39;</span><span class="p">,</span><span class="s">&#39;x==0&#39;</span><span class="p">);</span>
<span class="n">mfemPoisson</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span><span class="n">pde</span><span class="p">,</span><span class="n">option</span><span class="p">);</span>
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<pre>Uzawa-type MultiGrid Preconditioned PCG 
#dof:      336,  #nnz:     1200, V-cycle:  1, iter: 16,   err = 6.10e-09,   time = 0.44 s
Uzawa-type MultiGrid Preconditioned PCG 
#dof:     1312,  #nnz:     4832, V-cycle:  1, iter: 16,   err = 7.58e-09,   time =  0.1 s
Uzawa-type MultiGrid Preconditioned PCG 
#dof:     5184,  #nnz:    19392, V-cycle:  1, iter: 16,   err = 9.33e-09,   time = 0.17 s
Uzawa-type MultiGrid Preconditioned PCG 
#dof:    20608,  #nnz:    77696, V-cycle:  1, iter: 16,   err = 9.13e-09,   time = 0.51 s

 #Dof       h       ||u-u_h||    ||u_I-u_h||  ||sigma-sigma_h||||sigma-sigma_h||_{div}

  336   1.25e-01   1.29904e-01   3.24734e-02   1.00431e+00   1.01710e+01
 1312   6.25e-02   6.53343e-02   8.36322e-03   5.03316e-01   5.14701e+00
 5184   3.12e-02   3.27108e-02   2.10609e-03   2.51787e-01   2.58126e+00
20608   1.56e-02   1.63607e-02   5.27476e-04   1.25909e-01   1.29160e+00

 #Dof   Assemble     Solve      Error      Mesh    

  336   1.20e-01   4.40e-01   1.20e-01   2.00e-02
 1312   1.00e-01   1.00e-01   1.00e-01   1.00e-02
 5184   1.00e-01   1.70e-01   6.00e-02   1.00e-02
20608   7.00e-02   5.10e-01   1.20e-01   2.00e-02


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<h3 id="Pure-Neumann-boundary-condition">Pure Neumann boundary condition<a class="anchor-link" href="#Pure-Neumann-boundary-condition">&#182;</a></h3><p>In mixed formulation，the Neumann boundary condition for $d\nabla u$ becomes the Dirichlet boundary condiiton for $\sigma$. The space for $u$ is $L^2_0$ and thus one dof should be removed to have a non-singular system.</p>

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<div class=" highlight hl-matlab"><pre><span></span><span class="n">option</span><span class="p">.</span><span class="n">plotflag</span> <span class="p">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="n">mesh</span><span class="p">.</span><span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Neumann&#39;</span><span class="p">);</span>
<span class="n">mfemPoisson</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span><span class="n">pde</span><span class="p">,</span><span class="n">option</span><span class="p">);</span>
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<pre>Uzawa-type MultiGrid Preconditioned PCG 
#dof:      336,  #nnz:     1102, V-cycle:  1, iter: 16,   err = 3.20e-09,   time = 0.15 s
Uzawa-type MultiGrid Preconditioned PCG 
#dof:     1312,  #nnz:     4638, V-cycle:  1, iter: 17,   err = 3.67e-09,   time = 0.04 s
Uzawa-type MultiGrid Preconditioned PCG 
#dof:     5184,  #nnz:    19006, V-cycle:  1, iter: 18,   err = 3.25e-09,   time = 0.19 s
Uzawa-type MultiGrid Preconditioned PCG 
#dof:    20608,  #nnz:    76926, V-cycle:  1, iter: 18,   err = 8.29e-09,   time = 0.47 s

 #Dof       h       ||u-u_h||    ||u_I-u_h||  ||sigma-sigma_h||||sigma-sigma_h||_{div}

  336   1.25e-01   1.35274e-01   4.88517e-02   1.00659e+00   1.01710e+01
 1312   6.25e-02   6.55354e-02   1.00318e-02   5.03633e-01   5.14701e+00
 5184   3.12e-02   3.27194e-02   2.33376e-03   2.51827e-01   2.58126e+00
20608   1.56e-02   1.63613e-02   5.71884e-04   1.25914e-01   1.29160e+00

 #Dof   Assemble     Solve      Error      Mesh    

  336   6.00e-02   1.50e-01   3.00e-02   0.00e+00
 1312   1.00e-02   4.00e-02   2.00e-02   0.00e+00
 5184   2.00e-02   1.90e-01   4.00e-02   0.00e+00
20608   5.00e-02   4.70e-01   1.00e-01   1.00e-02


</pre>
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<h3 id="Pure-Dirichlet-boundary-condition">Pure Dirichlet boundary condition<a class="anchor-link" href="#Pure-Dirichlet-boundary-condition">&#182;</a></h3>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">%% Pure Dirichlet boundary condition.</span>
<span class="n">mesh</span><span class="p">.</span><span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Dirichlet&#39;</span><span class="p">);</span>
<span class="n">mfemPoisson</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span><span class="n">pde</span><span class="p">,</span><span class="n">option</span><span class="p">);</span>
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<pre>Uzawa-type MultiGrid Preconditioned PCG 
#dof:      336,  #nnz:     1232, V-cycle:  1, iter: 16,   err = 3.60e-09,   time = 0.09 s
Uzawa-type MultiGrid Preconditioned PCG 
#dof:     1312,  #nnz:     4896, V-cycle:  1, iter: 16,   err = 6.53e-09,   time = 0.05 s
Uzawa-type MultiGrid Preconditioned PCG 
#dof:     5184,  #nnz:    19520, V-cycle:  1, iter: 16,   err = 8.34e-09,   time = 0.16 s
Uzawa-type MultiGrid Preconditioned PCG 
#dof:    20608,  #nnz:    77952, V-cycle:  1, iter: 16,   err = 8.49e-09,   time = 0.43 s

 #Dof       h       ||u-u_h||    ||u_I-u_h||  ||sigma-sigma_h||||sigma-sigma_h||_{div}

  336   1.25e-01   1.29702e-01   3.08718e-02   1.00257e+00   1.01710e+01
 1312   6.25e-02   6.53059e-02   7.92226e-03   5.03081e-01   5.14701e+00
 5184   3.12e-02   3.27071e-02   1.99320e-03   2.51757e-01   2.58126e+00
20608   1.56e-02   1.63602e-02   4.99086e-04   1.25905e-01   1.29160e+00

 #Dof   Assemble     Solve      Error      Mesh    

  336   1.00e-02   9.00e-02   1.00e-02   0.00e+00
 1312   0.00e+00   5.00e-02   1.00e-02   0.00e+00
 5184   2.00e-02   1.60e-01   3.00e-02   1.00e-02
20608   5.00e-02   4.30e-01   1.00e-01   1.00e-02


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<h2 id="Conclusion">Conclusion<a class="anchor-link" href="#Conclusion">&#182;</a></h2><p>The optimal rates of convergence for $u$ and $\sigma$ are observed, namely, 1st order for L2 norm of u, L2 norm of $\sigma$ and H(div) norm of $\sigma$. The 2nd order convergent rates between two discrete functions $\|u_I - u_h\|$ and $\|\sigma_I - \sigma_h\|$ are known as superconvergence.</p>
<p>Triangular preconditioned GMRES (the default solver) and Uzawa preconditioned CG converges uniformly in all cases. Traingular preconditioner is two times faster than PCG although GMRES is used.</p>

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